Properties

Label 91.10.bb.a
Level $91$
Weight $10$
Character orbit 91.bb
Analytic conductor $46.868$
Analytic rank $0$
Dimension $328$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(328\)
Relative dimension: \(82\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9} + 87746 q^{11} + 320920 q^{14} - 19316 q^{15} + 9961468 q^{16} + 784580 q^{18} + 4117938 q^{19} - 2664378 q^{21} + 2436208 q^{22} - 13515348 q^{24} + 8735244 q^{26} + 597462 q^{28} - 17509544 q^{29} - 42948186 q^{31} + 7054324 q^{32} + 56872146 q^{33} + 3243616 q^{35} - 30560798 q^{37} + 57945680 q^{39} - 233246532 q^{40} - 6538696 q^{42} + 30777318 q^{44} - 118104 q^{45} - 221154484 q^{46} - 116058498 q^{47} + 67436008 q^{50} - 666401016 q^{52} + 193593000 q^{53} + 225058362 q^{54} - 208308092 q^{57} + 25517350 q^{58} + 432112842 q^{59} + 128172692 q^{60} + 495848688 q^{61} + 786184880 q^{63} + 553811470 q^{65} + 1418476692 q^{66} + 178582346 q^{67} + 135355380 q^{68} - 390093886 q^{70} - 791020032 q^{71} - 210348434 q^{72} + 45941286 q^{73} - 753352188 q^{74} - 2396110888 q^{78} - 1469125308 q^{79} + 2330725026 q^{80} - 5836467932 q^{81} + 511621342 q^{84} - 1231770100 q^{85} - 1420105854 q^{86} + 1924486176 q^{87} - 82536342 q^{89} - 839685044 q^{91} + 12912162192 q^{92} - 1527450306 q^{93} - 7925454132 q^{94} + 6855701310 q^{96} + 1571929618 q^{98} + 2719514056 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −11.4807 42.8465i −49.6096 + 28.6421i −1260.61 + 727.814i −39.7613 + 10.6540i 1796.77 + 1796.77i −6247.28 + 1151.15i 29597.7 + 29597.7i −8200.76 + 14204.1i 912.974 + 1581.32i
5.2 −10.9276 40.7824i 166.664 96.2237i −1100.38 + 635.307i 202.056 54.1409i −5745.47 5745.47i 1613.04 6144.24i 22648.2 + 22648.2i 8676.50 15028.1i −4415.99 7648.71i
5.3 −10.8870 40.6310i 120.266 69.4354i −1088.94 + 628.702i −348.043 + 93.2579i −4130.57 4130.57i 4183.56 + 4780.32i 22171.2 + 22171.2i −198.943 + 344.579i 7578.32 + 13126.0i
5.4 −10.8713 40.5723i −111.169 + 64.1836i −1084.52 + 626.147i −1911.08 + 512.072i 3812.63 + 3812.63i 3916.23 5001.68i 21987.4 + 21987.4i −1602.44 + 2775.51i 41551.8 + 71969.9i
5.5 −10.6545 39.7631i −221.572 + 127.925i −1024.18 + 591.312i 1246.07 333.883i 7447.41 + 7447.41i 4766.00 + 4199.86i 19520.9 + 19520.9i 22887.9 39642.9i −26552.5 45990.2i
5.6 −10.1912 38.0341i −54.7649 + 31.6185i −899.330 + 519.228i −1311.11 + 351.311i 1760.70 + 1760.70i 4438.33 + 4544.76i 14658.1 + 14658.1i −7842.04 + 13582.8i 26723.6 + 46286.7i
5.7 −10.1645 37.9344i 9.27740 5.35631i −892.300 + 515.170i 2307.23 618.220i −297.489 297.489i 1373.49 6202.19i 14394.3 + 14394.3i −9784.12 + 16946.6i −46903.7 81239.5i
5.8 −10.1089 37.7268i −116.736 + 67.3977i −877.720 + 506.752i 1557.93 417.446i 3722.77 + 3722.77i −5979.50 + 2144.56i 13850.5 + 13850.5i −756.597 + 1310.46i −31497.9 54555.9i
5.9 −9.88529 36.8924i 230.291 132.958i −819.927 + 473.385i −1552.67 + 416.037i −7181.65 7181.65i −5367.23 + 3398.00i 11741.9 + 11741.9i 25514.4 44192.2i 30697.2 + 53169.1i
5.10 −9.84620 36.7465i 121.299 70.0318i −809.953 + 467.627i 1878.59 503.366i −3767.76 3767.76i −468.182 + 6335.17i 11385.7 + 11385.7i −32.5834 + 56.4360i −36993.9 64075.2i
5.11 −9.41397 35.1334i 55.9682 32.3132i −702.328 + 405.489i −1552.01 + 415.860i −1662.16 1662.16i −5033.95 3874.65i 7689.56 + 7689.56i −7753.21 + 13429.0i 29221.2 + 50612.6i
5.12 −9.08088 33.8903i −163.524 + 94.4107i −622.686 + 359.508i 197.881 53.0221i 4684.55 + 4684.55i 1215.08 6235.16i 5135.93 + 5135.93i 7985.28 13830.9i −3593.87 6224.77i
5.13 −8.94964 33.4005i −222.806 + 128.637i −592.093 + 341.845i −2577.17 + 690.551i 6290.58 + 6290.58i −4593.27 + 4388.10i 4197.96 + 4197.96i 23253.6 40276.4i 46129.5 + 79898.7i
5.14 −8.43697 31.4872i −61.0517 + 35.2482i −476.857 + 275.313i 1307.03 350.218i 1624.96 + 1624.96i 6319.13 + 649.753i 890.360 + 890.360i −7356.63 + 12742.1i −22054.8 38200.0i
5.15 −7.84691 29.2851i 148.521 85.7485i −352.636 + 203.594i 1538.27 412.177i −3676.58 3676.58i −6345.80 + 290.604i −2246.96 2246.96i 4864.12 8424.90i −24141.2 41813.9i
5.16 −7.81263 29.1571i −10.1416 + 5.85528i −345.695 + 199.587i −1154.98 + 309.476i 249.956 + 249.956i −2271.29 + 5932.53i −2408.20 2408.20i −9772.93 + 16927.2i 18046.8 + 31258.0i
5.17 −7.64658 28.5374i 145.433 83.9657i −312.510 + 180.428i −2483.48 + 665.446i −3508.23 3508.23i 6197.43 1394.80i −3157.54 3157.54i 4258.97 7376.76i 37980.2 + 65783.7i
5.18 −7.42068 27.6944i 70.6230 40.7742i −268.507 + 155.022i −828.537 + 222.006i −1653.29 1653.29i −4105.26 4847.72i −4094.38 4094.38i −6516.43 + 11286.8i 12296.6 + 21298.4i
5.19 −7.37110 27.5093i 45.1180 26.0489i −259.024 + 149.548i 378.159 101.327i −1049.16 1049.16i 6292.42 871.216i −4287.52 4287.52i −8484.41 + 14695.4i −5574.89 9655.99i
5.20 −7.27837 27.1633i −177.038 + 102.213i −241.463 + 139.408i 171.168 45.8642i 4064.99 + 4064.99i −5809.82 2568.98i −4636.82 4636.82i 11053.5 19145.3i −2491.64 4315.65i
See next 80 embeddings (of 328 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.bb.a 328
7.d odd 6 1 inner 91.10.bb.a 328
13.d odd 4 1 inner 91.10.bb.a 328
91.bb even 12 1 inner 91.10.bb.a 328
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.bb.a 328 1.a even 1 1 trivial
91.10.bb.a 328 7.d odd 6 1 inner
91.10.bb.a 328 13.d odd 4 1 inner
91.10.bb.a 328 91.bb even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).